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    <title>GROOPS - KalmanFilter</title>

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            <h1>KalmanFilter</h1><p>

The program computes time variable gravity fields using the Kalman filter approach of</p><p>Kurtenbach, E., Eicker, A., Mayer-Gürr, T., Holschneider, M., Hayn, M., Fuhrmann, M., and Kusche, J. (2012).
Improved daily GRACE gravity field solutions using a Kalman smoother. Journal of Geodynamics, 59–60, 39–48.
<a href="https://doi.org/10.1016/j.jog.2012.02.006" target="_blank">https://doi.org/10.1016/j.jog.2012.02.006</a>.</p><p>The updated state $\mathbf{x}_t^+$ is determined by solving the least squares adjustment
\[
\mathbf{l}_t = \mathbf{A}_t \mathbf{x}_t + \mathbf{e}_t \hspace{25pt} \mathbf{e}_t \sim \mathcal{N}(0, \mathbf{R}_t)\\
\mathbf{B} \mathbf{x}^+_{t-1} = \mathbf{I} \mathbf{x}_t + \mathbf{v}_t\hspace{25pt} \mathbf{v} \sim \mathcal{N}(0,\mathbf{Q} + \mathbf{B} \mathbf{P}^+_{t-1}\mathbf{B}^T).
\]In normal equation form this can be written as
\[
\hat{\mathbf{x}}_t = \mathbf{x}^+_t = (\mathbf{N}_t + \mathbf{P}^{-^{-1}}_t)^{-1}(\mathbf{n}_t + \mathbf{P}^{-^{-1}}_t \mathbf{x}^-_t),
\]where $\mathbf{x}_t^- = \mathbf{B} \mathbf{x}^+_{t-1}$ and $\mathbf{P}_t^{-} = \mathbf{Q} + \mathbf{B} \mathbf{P}^+_{t-1}\mathbf{B}^T$
are the predicted state and its covariance matrix.</p><p>The process dynamic $\mathbf{B}, \mathbf{Q}$ is represented as an <a class="groops-ref" href="fundamentals.autoregressiveModel.html">autoregressive model</a>,
and passed to the program through <a class="groops-class" href="fileFormat_matrix.html">inputfileAutoregressiveModel</a>.
The sequence of normal equations $\mathbf{N}_t, \mathbf{n}_t$ are given as list of <a class="groops-class" href="fileFormat_normalEquation.html">inputfileNormalEquations</a>,
which can be generated using <a class="groops-class" href="loopType.html">loops</a>.
In the same way, the <a class="groops-file" href="fileFormat_matrix.html">matrix files</a> for <strong class="groops-config-element">outputfileUpdatedState</strong> and <strong class="groops-config-element">inputfileUpdatedStateCovariance</strong>
can also be specified using <a class="groops-class" href="loopType.html">loops</a>.</p><p>If no <a class="groops-class" href="fileFormat_matrix.html">inputfileInitialState</a> is set, a zero vector with appropriate dimensions is used.
The <a class="groops-class" href="fileFormat_matrix.html">inputfileInitialStateCovarianceMatrix</a> however must be given.</p><p>See also <a class="groops-program" href="KalmanBuildNormals.html">KalmanBuildNormals</a>, <a class="groops-program" href="KalmanSmoother.html">KalmanSmoother</a>.
</p>
<table class="table table-hover">
<tr class="table-primary"><th>Name</th><th>Type</th><th>Annotation</th></tr>
<tr class="table-light"><td class="m-0"><div class="h-100 config-tree depth-0"><div class="h-100 config mustset-unbounded">outputfileUpdatedState</div></div></td><td>filename</td><td>estimated state x+ (nx1-matrix)</td></tr>
<tr class=""><td class="m-0"><div class="h-100 config-tree depth-0"><div class="h-100 config optional-unbounded">outputfileUpdatedStateCovarianceMatrix</div></div></td><td>filename</td><td>estimated state' s covariance matrix Cov(x+)</td></tr>
<tr class="table-light"><td class="m-0"><div class="h-100 config-tree depth-0"><div class="h-100 config mustset-unbounded">inputfileNormalEquations</div></div></td><td>filename</td><td>normal equations input file</td></tr>
<tr class=""><td class="m-0"><div class="h-100 config-tree depth-0"><div class="h-100 config optional">inputfileInitialState</div></div></td><td>filename</td><td>initial state x0</td></tr>
<tr class="table-light"><td class="m-0"><div class="h-100 config-tree depth-0"><div class="h-100 config mustset">inputfileInitialStateCovarianceMatrix</div></div></td><td>filename</td><td>initial state's covariance matrix Cov(x0)</td></tr>
<tr class=""><td class="m-0"><div class="h-100 config-tree depth-0"><div class="h-100 config mustset">inputfileAutoregressiveModel</div></div></td><td>filename</td><td>file name of autoregressive model</td></tr>
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